Quasi-hereditary algebras and the category of modules with standard filtration

OF MODULES WITH STANDARD FILTRATION

DAG OSKAR MADSEN

Abstract. This paper is a survey on some of the most basic results in the theory of quasi-hereditary algebras. In the last section we briefly discuss a recent development.

This survey covers some of the basic results in the theory of quasi-hereditary algebras. It corresponds to the first half of the talk given by the author at the CIMPA Research School on “Algebraic and Geometric Aspects of Rep- resentation Theory”, Curitiba, Brazil, February 25 to March 9 2013.

Quasi-hereditary algebras appear in the representation theory of algebraic groups and semi-simple Lie algebras [Par], and recently also in algebraic geometry [HP]. Quasi-hereditary algebras were originally defined by Cline, Parshall and Scott [CPS], the definition first appearing in print in [Sco]. The general ideas were anticipated by other authors in earlier works like [Nic]

and [BGG]. In these notes we follow more closely the approach of Dlab and Ringel [DR1]. The presentation borrows from the survey papers [Par] and [DR2]. The interested reader should consult these sources for a treatment of more advanced topics, including the important concept of characteristic tilting module which is not treated here.

In the last section we briefly discuss a recent development in the theory.

1. Standard and costandard modules

Let k be a field and let B be a finite-dimensional k-algebra. Through- out J denotes the Jacobson radical of the algebra B. The category of left B-modules is denoted by ModB and the full subcategory of finitely gener- ated B-modules is denoted by modB. Fix an ordering on a complete set of non-isomorphic simple B-modules S1, . . . , Sr. Let P1, . . . , Pr denote the corresponding indecomposable projective modules andI₁, . . . , I_r denote the corresponding indecomposable injective modules.

Definition 1.1. For each 1≤i≤r, define thestandard module ∆_ito be the largest quotient of the projective module Pi having no simple composition factors S_j with j > i. Dually, define the costandard module ∇_i to be the largest submodule of the injective module Ii having no simple composition factorsS_j with j > i. Let

∆ =

r

M

i=1

∆_i

1

and ∇ = Lr

i=1∇_i. (Note: In some papers ∆ and ∇ are used to denote the sets of standard and costandard modules respectively rather than their respective direct sums.)

The standard and costandard modules depend on the chosen ordering of the simple modules, a different choice of ordering may give different standard and costandard modules.

Example 1.2. LetB be the path algebraB =kQ/I, whereQis the quiver Q: •₁

α ))•₂

δ

ii

β ))•₃

γ

ii

and I is the ideal

I =hβα, βγ, δγ, αδ−γβi.

In path algebra examples we let the ordering of simple modules coincide with the indexing of vertices in the quiver.

The projective modules are:

P₁: S1

S₂ S1

P₂: S2

S₁ S₃ S2

P₃: S3

S₂

The standard modules are:

∆₁: S1 ∆₂: S2

S1

∆₃: S3

S2

The injective modules are:

I1: S1

S2

S1

I2: S2

S1 S3

S2

I3: S2

S3

The costandard modules are:

∇₁: S1 ∇₂: S1

S2

∇₃: S2

S3.

The duality D = Hom_k(−,k) : modB → modB^op sends costandard B- modules to standard modules over the opposite algebraB^op. By this duality, for any statement about standard modules, there is a corresponding state- ment about costandard modules.

Without any further assumptions, we have the following vanishing result.

Lemma 1.3. Ext¹_B(∆,∇) = 0.

Proof. Suppose Ext¹_B(∆i,∇_j) 6= 0 for some 1 ≤ i, j ≤ r. By applying the functor Hom_B(−,∇_j) to the exact sequence

0→Ω(∆_i)→P_i→∆_i→0, we get a long-exact sequence

0→HomB(∆i,∇_j)→HomB(Pi,∇_j)→HomB(Ω(∆i),∇_j)→Ext¹_B(∆i,∇_j)→0.

Since Ext¹_B(∆i,∇_j)6= 0, we have Hom_B(Ω(∆i),∇_j)6= 0. Therefore the top of Ω(∆_i) must have a composition factor S_l with l ≤ j. By the definition of ∆i, the composition factors of the top of Ω(∆i) must have index greater thani. Soi < l ≤j. By duality, from the exact sequence

0→ ∇_j →I_j →Ω⁻¹(∇_j)→0,

we obtain that the socle of Ω⁻¹(∇_j) must have a composition factorSv with j < v ≤ i. Since i < j and j < i, we reach a contradiction. In conclusion

Ext¹_B(∆,∇) = 0.

Let M be a finitely generated B-module. We say that M admits a ∆- filtration if there is a filtration 0 = M0 ⊆ M1 ⊆. . . ⊆Mt =B where the subfactors M_j/Mj−1 are standard modules for all and 1 ≤ j ≤ t. If M admits a ∆-filtration, then it follows from Lemma 1.3 and induction on the length of the ∆-filtration that Ext¹_B(M,∇) = 0.

Definition 1.4. We say thatB is aquasi-hereditary algebra if (i)B admits a ∆-filtration, and (ii) EndB(∆i) is a division ring for all 1≤i≤r.

The lack of left-right symmetry in this definition is only apparent. We first deal with the second condition.

Lemma 1.5. For each 1≤ i≤r, the algebra End_B(∆_i) is a division ring if and only if EndB(∇_i) is a division ring.

Proof. Suppose EndB(∇_i) is not a division ring and let f:∇_i → ∇_i be a non-zero non-isomorphism. Let g:P_i → ∇_i be a map with img= soc(∇_i).

Since img ⊆ imf and Pi is projective, there is a map h:Pi → ∇_i with g=f h. All composition factors of ∇_i have index less than or equal to i, so the map hfactors through the surjection j:Pi →∆i.

P_i

j

vv ^h}} ^g

∆_i

t //∇_i

f //∇_i.

Since soc(∇_i)⊆kerf, the image ofh is strictly larger than soc(∇_i). There- fore the image oft: ∆_i→ ∇_i is strictly larger than soc(∇_i).

Since img ⊆ imt and Pi is projective, there is a map h⁰:Pi → ∆i with g=th⁰. The map h⁰ factors through the surjectionj:P_i →∆_i.

Pi j

vv ^h0}} ^g

∆_i

f⁰

//∆_i

t //∇_i.

The image of t is strictly larger than soc(∇_i), so h⁰ is not an epimor- phism. Thereforef⁰: ∆_i →∆_i is a non-zero non-isomorphism, which means End_B(∆_i) is not a division ring. So if End_B(∆_i) is a division ring, then EndB(∇_i) is a division ring. By duality, the statement of the lemma fol-

lows.

If End_B(∆_i) is a division ring, then the composition factors of J∆_i have index strictly less thani, and, as a consequence of the lemma, all composition factors of ∇_i/soc(∇_i) have index strictly less thani.

In the definition of quasi-hereditary algebras, the existence of filtrations can be replaced by a self-dual statement, namely the vanishing of Ext²_B(∆,∇).

Theorem 1.6. The algebra B is quasi-hereditary if and only if (i) Ext²_B(∆,∇) = 0, and

(ii) End_B(∆_i) is a division ring for all 1≤i≤r.

Proof. SupposeB is quasi-hereditary. Then each indecomposable projective moduleP_i admits a ∆-filtration. SinceP_i has a simple top, the top quotient in the filtration must be ∆i. Therefore the kernel Ω(∆i) of the morphism Pi → ∆i admits a ∆-filtration. So Ext¹_B(Ω(∆i),∇) = 0 by the remark following Lemma 1.3. By dimension shift

Ext²_B(∆i,∇)'Ext¹_B(Ω(∆i),∇) = 0.

Since this is true for all 1≤i≤r, we get Ext²_B(∆,∇) = 0.

For the converse, assume End_B(∆i) is a division ring for all 1≤i≤r.

LetM be a finitely generatedB-module. Under the hypothesis Ext²_B(∆,∇) = 0, we claim thatM admits a ∆-filtration if and only if Ext¹_B(M,∇) = 0.

IfM admits a ∆-filtration, then Ext¹_B(M,∇) = 0 as already remarked.

Assume Ext¹_B(M,∇) = 0. Let l be the smallest index such that Sl is a composition factor of top(M). Supposej≤l. There is a long-exact sequence

0→HomB(M, Sj)→HomB(M,∇_j)→HomB(M,∇_j/soc(∇_j))

→Ext¹_B(M, S_j)→Ext¹_B(M,∇_j)→. . .

Since all composition factors of ∇_j/soc(∇_j) have index strictly less than j, we have Hom_B(M,∇_j/soc(∇_j)) = 0; Since Hom_B(M,∇_j/soc(∇_j)) = 0 and Ext¹_B(M,∇_j) = 0, we have Ext¹_B(M, S_j) = 0. There is a long-exact sequence

0→Hom_B(M, J∆_l)→Hom_B(M,∆_l)→Hom_B(M, S_l)→Ext¹_B(M, J∆_l)→. . .

Since all composition factors ofJ∆lhave index less thanl, we have HomB(M, J∆l) = 0 and also Ext¹_B(M, J∆_l) = 0 by induction on the length ofJ∆_l. Any mor-

phism M → S_l will therefore factor through the map ∆_l → S_l. So there exists a surjective morphism from M to ∆_l. Let f:M →∆_l denote such a morphism with kernelM⁰. There is a long-exact sequence

0→Hom_B(∆_l,∇)→Hom_B(M,∇)→Hom_B(M⁰,∇)

→Ext¹_B(∆l,∇)→Ext¹_B(M,∇)→Ext¹_B(M⁰,∇)→Ext²_B(∆l,∇)→. . . By assumption Ext¹_B(M,∇) = 0 and Ext²_B(∆_l,∇) = 0, so Ext¹_B(M⁰,∇) = 0.

The claim follows by induction on the length ofM.

We have Ext¹_B(B,∇) = 0, so if Ext²_B(∆,∇) = 0, then B admits a ∆- filtration and therefore B is quasi-hereditary.

Corollary 1.7. The algebra B is quasi-hereditary if and only if B^op is quasi-hereditary.

Proof. Condition (i) in Theorem 1.6 is a self-dual statement. By Lemma 1.5, condition (ii) in Theorem 1.6 holds if and only if the dual statement

holds.

Since the conditions only depend on the module category, Theorem 1.6 also shows that the property of being quasi-hereditary is Morita invariant.

2. Homological properties

Throughout this sectionB denotes some quasi-hereditary algebra.

Theorem 2.1. Extⁿ_B(∆,∇) = 0 for alln >0.

Proof. We know Ext¹_B(∆,∇) = 0 by Lemma 1.3. Assume as induction hypothesis that Ext^k_B(∆,∇) = 0 for a givenk >0. For any 1≤i≤r, since Ω(∆i) admits a ∆-filtration, we have Ext^k_B(Ω(∆i),∇) = 0. By dimension shift

Ext^k+1_B (∆i,∇)'Ext^k_B(Ω(∆i),∇) = 0.

Therefore Ext^k+1_B (∆,∇) = 0. The statement of the theorem follows.

Theorem 2.2. HomB(∆i,∆j) = 0 for all 1≤j < i≤r.

Extⁿ_B(∆i,∆j) = 0 for alln >0 and 1≤j≤i≤r.

Proof. The composition factors of ∆j have index strictly less than j, so Hom_B(∆_i,∆_j) = 0 for all 1≤j < i≤r.

Let 1 ≤ i ≤ r. For any j ≤ i, we have HomB(∆i,∇_j/soc(∇_j)) = 0 since all composition factors of ∇_j/soc(∇_j) have index strictly less than j.

Consider the following part of the relevant long-exact sequence.

. . .→Hom_B(∆_i,∇_j/soc(∇_j))→Ext¹_B(∆_i, S_j)→Ext¹_B(∆_i,∇_j)→. . .

Since Hom_B(∆_i,∇_j/soc(∇_j)) = 0 and Ext¹_B(∆_i,∇_j) = 0, we have Ext¹_B(∆_i, S_j) = 0 whenever j≤i.

Assume as induction hypothesis that Ext^k_B(∆i, Sj) = 0 for all j ≤ i.

Since all composition factors of ∇_j/soc(∇_j) have index strictly less than j, it follows that Ext^k_B(∆_i,∇_j/soc(∇_j)) = 0 for all j ≤ i. Consider the following part of the long-exact sequence.

. . .→Ext^k_B(∆i,∇_j)→Ext^k_B(∆i,∇_j/soc(∇_j))→Ext^k+1_B (∆i, Sj)→Ext^k+1_B (∆i,∇_j)→. . . Here the end terms are zero, so

Ext^k+1_B (∆_i, S_j)'Ext^k_B(∆_i,∇_j/soc(∇_j)) = 0

for all j≤i. Hence Extⁿ_B(∆_i, S_j) = 0 for all n >0 and 1≤j≤i≤r.

Since all composition factors of ∆j have index j or less, it follows by induction on the length of ∆_j that Extⁿ_B(∆_i,∆_j) = 0 for all n > 0 and

1≤j≤i≤r.

One consequence of Ext¹_B(∆i,∆j) = 0 for all j ≤ i is that if a mod- ule M admits a ∆-filtration, then the ∆-factors can always be chosen in non-increasing order, meaning that the standard modules with highest in- dex appear at the bottom of the filtration while he standard modules with smallest index appear at the top of the filtration.

We now look at bounds for homological dimensions. For more on projec- tive dimensions in exact sequences, see section 1 of [Mad].

Lemma 2.3. Let R be a ring and0→L→M →N →0 an exact sequence of R-modules. Then

(a) pd_RM ≤max{pd_RL,pd_RN}.

(b) If pd_RM <max{pd_RL,pd_RN}, then pd_RL+ 1 = pd_RN. Proof. (a) The exact sequence of functors

Extⁿ_R(N,−)→Extⁿ_R(M,−)→Extⁿ_R(L,−)

shows that pd_RM ≥ n implies [pd_RL ≥ n or pd_RN ≥ n]. Therefore pd_RM ≤max{pd_RL,pd_RN}.

(b) Assume pd_RM = n < ∞. Then there is an epimorphism of func- tors Extⁿ_R(L,−) → Extⁿ⁺¹_R (N,−) → 0 and isomorphisms Ext^m_R(L,−) ' Ext^m+1_R (N,−) for m > n. So if pd_RN > n, then pd_RL = pd_RN −1.

Similarly if pd_RL > n, then pd_RN = pd_RL+ 1.

Theorem 2.4. pd_B(∆i)≤r−i.

Proof. The standard module ∆_r is projective, so pd_B(∆_r) = pd_B(P_r) = 0.

Let 1≤n < r and assume pd_B(∆i) ≤r−i whenever i > n. Consider the exact sequence

0→Ω(∆_n)→P_n→∆_n→0.

The quotients in the ∆-filtration of Ω(∆_n) have index strictly greater than n, so by repeated use of Lemma 2.3(a) we get pd_B(Ω(∆_n))≤r−n−1. Hence pd_B(∆n) = pd_B(Ω(∆n)) + 1≤r−n. The theorem follows by induction.

Theorem 2.5. gldimB ≤2r−2.

Proof. The standard module ∆1 is simple, so pd_B(S1) = pd_B(∆1)≤r−1 = r+ 1−2 by the previous theorem. Let 1< n ≤ r and assume pd_B(S_i) ≤ r+i−2 whenever i < n. Consider the exact sequence

0→J∆n→∆n→Sn→0.

The composition factors of J∆n have index strictly less than n, so by re- peated use of Lemma 2.3(a) we get pd_B(J∆_n)≤r+n−3. Also by Lemma 2.3(a) we have

pd_B(∆n)≤max{pd_B(J∆n),pd_B(Sn)}.

If pd_B(∆_n) = max{pd_B(J∆_n),pd_B(S_n)}, then by Theorem 2.4 pd_B(S_n)≤pd_B(∆_n)≤r−n.

If pd_B(∆_i)<max{pd_B(J∆_n),pd_B(S_n)}, then by Lemma 2.3(b) pd_B(Sn) = pd_B(J∆n) + 1≤r+n−2.

In both cases pd_B(Sn) ≤ r+n−2. By induction pd_B(Si) ≤r+i−2 for 1≤i≤r.

Since gldimB = max{pd_B(Si)|1≤i≤r}, we get gldimB≤r+r−2 = 2r−2.

Lemma 2.6. Let Abe a finite-dimensionalk-algebra ande∈Aan idempo- tent. Suppose the two-sided ideal AeA is projective as a left A-module. Let A¯ denote the algebra A¯=A/AeA. Then

Extⁿ_A¯(M, N)'Extⁿ_A(M, N) for all n≥0 and all A-modules¯ M, N.

Proof. There is a full embedding of module categoriesModA¯→ModAthat we exploit throughout the proof. The result forn= 0 is immediate, we have

HomA¯(M, N)'Hom_A(M, N) for all ¯A-modules M, N.

Let N be an ¯A-module. For any f ∈ HomA(AeA, N) and aea⁰ ∈ AeA, we havef(aea⁰) =ae·f(ea⁰) = 0, so Hom_A(AeA, N) = 0. Using that AeA is projective, we see from the long-exact sequence obtained by applying HomA(−, N) to the exact sequence 0→AeA→A→A/AeA→0 that

Extⁿ_A(A/AeA, N) = 0

for all n >0. It follows that Extⁿ_A( ¯P , N) = 0 for any projective ¯A-module P¯ and n >0.

Let M be an ¯A-module and 0→ ΩA¯(M)→ PA¯(M) →M → 0 an exact sequence with PA¯(M) projective as ¯A-module. There is a commutative

diagram with exact rows HomA¯(PA¯(M), N) ^//

o

HomA¯(ΩA¯M, N) ^//

o

Ext¹_A¯(M, N) ^//

0

HomA(PA¯(M), N) ^//HomA(ΩA¯M, N) ^//Ext¹_A(M, N) ^//0.

Since the two left-most downward arrows are isomorphisms, we get Ext¹_A¯(M, N)'Ext¹_A(M, N)

for all ¯A-modules M, N.

We proceed by induction. Assume there is ann≥1 such that Extⁿ_A¯(M, N)' Extⁿ_A(M, N) for all ¯A-modulesM, N. There is a commutative diagram

0 ^//Extⁿ_A¯(ΩA¯M, N) ^{∼ //}

o

Extⁿ⁺¹_A¯ (M, N) ^//

0

0 ^//Extⁿ_A(ΩA¯M, N) ^{∼ //}Extⁿ⁺¹_A (M, N) ^//0.

By the induction hypothesis, the left downward arrow is an isomorphism. It follows that Extⁿ⁺¹_A¯ (M, N) 'Extⁿ⁺¹_A (M, N) for all ¯A-modules M, N. The

lemma follows by induction.

Let ¯Bdenote the algebra ¯B =B/BeB, whereeis a primitive idempotent corresponding to the simple B-module with highest index Sr. The algebra B¯ has standard modules ¯∆_i = ∆_i and costandard modules ¯∇_i = ∇_i for 1≤i≤r−1.

Theorem 2.7. The algebra B¯ is quasi-hereditary.

Proof. Let 1B=e1+. . .+esbe a decomposition of the identity into primitive orthogonal idempotents. SinceBis quasi-hereditary, every finitely generated projective B-module P admits a ∆-filtration. By the comment following Theorem 2.2, any non-zero morphismg: ∆r=Be→Pmust be an inclusion.

The two-sided ideal BeB considered as a left B-module has a decompo- sition

BeB'

s

M

i=1

BeBe_i.

For each primitive idempotent f, we have a further decomposition BeBf '

t

M

j=1

Bqj,

where {q₁, . . . , q_t} is a k-basis for eBf. For each 1 ≤j ≤t, there is a sur- jectionhj:Be→Bqj given by right multiplication by qj. The composition Be −→^hj Bq_j ,→ Bf is an inclusion, so each h_j must be an isomorphism.

ThereforeBeBas a leftB-module is a direct sum of copies ofBeand hence it is projective.

Let ¯∆ =Lr−1

i=1 ∆_i. From Lemma 2.6 we get

Ext²B¯( ¯∆,∇)¯ 'Ext²_B( ¯∆,∇) = 0.¯ For 1≤i≤r−1 we also get

EndB¯( ¯∆i)'EndB( ¯∆i)'EndB(∆i),

which is a division ring. By Theorem 1.6, the algebra ¯B is quasi-hereditary.

3. Examples of quasi-hereditary algebras

The question whether an algebra is quasi-hereditary or not might depend on the chosen ordering of simple modules. Directed algebras are quasi- hereditary in (at least) two different ways, with simple standard modules or with projective standard modules. An algebraB is calleddirected if there is an ordering on a complete set of non-isomorphic indecomposable projective modulesP₁, . . . , P_rsuch that Hom_B(P_i, P_j) = 0 whenever 1≤i < j ≤r and EndB(Pi) is a division ring for all 1≤i≤r. Examples of directed algebras are path algebras kQ/I with Q a directed quiver. Let B be a directed algebra. If the same ordering S1, . . . , Sr is used for the simple modules, then ∆_i = S_i for all 1 ≤i ≤ r and the algebra is quasi-hereditary. If the opposite ordering is used, then ∆_i =P_i for all 1≤i≤r and the algebra is quasi-hereditary.

Semi-simple algebras are quasi-hereditary for any ordering of simple mod- ules. Hereditary algebras also have this property.

Theorem 3.1. Hereditary algebras are quasi-hereditary for any ordering of simple modules.

Proof. Let B be an hereditary algebra and fix an ordering of the simple modules. IfP is a non-zero finitely generated projective module, then there is a surjectionf:P →∆_i onto a standard module ∆_i. If kerf = 0, then P admits a trivial ∆-filtration. SinceB is hereditary, the kernel kerf is either zero or a non-zero projective module of shorter length thanP. By induction on the length of P, every finitely generated projectiveB-moduleP admits a ∆-filtration. In particularB admits a ∆-filtration.

Let h: ∆i → ∆i be a non-zero non-isomorphism for some 1 ≤ i ≤ r.

It lifts to a non-zero non-isomorphism ˜h: P_i → P_i. Since B is hereditary, such a morphism cannot exist. Therefore EndB(∆i) is a division ring for all 1≤i≤r. This proves that B is quasi-hereditary.

So all algebrasBwith gldimB ≤1 are quasi-hereditary. Global dimension two algebras are also quasi-hereditary, but in this case we have to be careful with the choice of ordering.

Theorem 3.2. Algebras with global dimension two are quasi-hereditary.

Proof. LetB be an algebra of global dimension two. Without loss of gener- ality we can assumeBis basic. The crucial property of global dimension two algebras is that the kernel of a map between projective modules is projec- tive. Let 1_B=f1+. . .+fr be a decomposition of the identity into primitive orthogonal idempotents. Choose a primitive idempotentefrom this decom- position such that the projective module Be has minimal Loewy length.

Since Be has minimal Loewy length, any non-zero morphism g: Be → P with P projective must be an inclusion. In particular, any non-zero mor- phismBe→Bemust be an isomorphism. Therefore End_B(Be) is a division ring.

Since any non-zero morphismg:Be→P is an inclusion, in the same way as in the proof of Theorem 2.7 it follows that BeB is projective as a left B-module. More preciselyBeB as a leftB-module is a direct sum of copies of Be.

To prove that B is quasi-hereditary we use an inductive argument. If r = 1, then B is local of finite global dimension, hence semi-simple and therefore quasi-hereditary. If r >1, choose the simple module correspond- ing to the primitive idempotent e to be maximal in the ordering. Then

∆_r = Be is a projective standard module and BeB admits a standard fil- tration. Consider the basic algebra ¯B = B/BeB. It has r −1 primitive orthogonal idempotents. From Lemma 2.6 it follows that gldim ¯B ≤2. The full embedding modB¯ → modB sends standard ¯B-modules to standardB- modules. Assume ¯B is quasi-hereditary. Since both ¯B and BeB admit

∆-filtrations, also B admits a ∆-filtration. By assumption End_B(∆_i) is a division ring for all 1≤i≤r−1. Also End_B(∆r) = End_B(Be) is a division ring. So ¯B being quasi-hereditary implies that B is quasi-hereditary. By induction onr, the algebra B is quasi-hereditary.

In the next example we consider a global dimension two path algebra with a quiver that is not directed.

Example 3.3. LetB be the path algebraB =kQ/I, whereQis the quiver Q: •_a

α ))•_b

β

•_d

δ

II

•_c

γ

jj

and I is the ideal

I =hβαi.

This algebra has global dimension two. The indecomposable projective mod- ule with shortest Loewy length isPa, so we let S4 =Sa.

The algebra ¯B =B/Be_aB is isomorphic to the path algebra of the quiver Q¯: •_b ^{β //}•_c ^{γ //}•_d

Since ¯B is hereditary, any ordering will do, so let for instance S3 = Sd, S₂ =S_c,S₁ =S_b. With this ordering B is quasi-hereditary.

As a consequence of the previous theorem, ifB is an algebra of finite rep- resentation type with indecomposable modules M₁, . . . , M_s, then the Aus- lander algebra [EndB(⊕^s_i=1Mi)]^op is quasi-hereditary. For a proof that Aus- lander algebras have global dimension at most two, see [Aus].

By Theorem 2.5, quasi-hereditary algebras have finite global dimension.

Not all algebras with finite global dimension are quasi-hereditary, as the following example shows. This example first appeared in [Gre].

Example 3.4. LetB be the path algebraB =kQ/I, whereQis the quiver

Q: •_a

α1

))

α2

•_b

β1

jj

β2

WW

and I is the ideal

I =hβ₁α1, β2α1, β2α2, α2β1i.

This algebra has global dimension four. The bound gldimB ≤ 2r−2 = 2 from Theorem 2.5 is not satisfied, soB is not quasi-hereditary.

As an illustration that quasi-hereditary algebras are ubiquitous, we record the following important theorem by Iyama [Iya].

Theorem 3.5. Let B be a finite dimensional k-algebra and let M be a finitely generatedB-module. Then there exists a finitely generatedB-module X such that[EndB(M⊕X)]^op is quasi-hereditary.

4. The category F(∆)

Suppose B is a quasi-hereditary algebra. Let F(∆) denote the full sub- category of modB consisting of the B-modules which admit a ∆-filtration.

The category F(∆) is a resolving subcategory of modB, that is, (i) it contains the indecomposable projective B-modules, (ii) it is closed under extensions and direct summands, and (iii) it is closed under kernels of epi- morphisms. The most convenient way to prove this fact is to use the de- scription of F(∆) given in the proof of Theorem 1.6; a finitely generated B-module M is in F(∆) if and only if Ext¹_B(M,∇) = 0. Condition (i) is then obvious. Also, condition (ii) clearly holds. To prove condition (iii), let 0→L→M →N →0 be an exact sequence of finitely generatedB-modules withM and N inF(∆). In the long-exact sequence

. . .→Ext¹_B(M,∇)→Ext¹_B(L,∇)→Ext²_B(N,∇)→. . .

we have Ext¹_B(M,∇) = 0 by assumption. Since N is in F(∆), it follows from Theorem 2.1 that Ext²_B(N,∇) = 0. Therefore Ext¹_B(L,∇) = 0, andN is in F(∆). So F(∆) is closed under kernels of epimorphisms.

The category F(∆) is not an abelian subcategory of modB, unless all standard modules are simple. The categoryF(∆) does have its own kernels and cokernels, but this is a subject we will not go into here.

In the next section we come back to a more subtle property of F(∆); it has Auslander-Reiten sequences.

5. Box characterization of quasi-hereditary algebras One way to read Theorem 2.2 is that a quasi-hereditary structure on an algebra imposes a certain directedness to the algebra and its module category. This theme can be taken much further, and a characterization of quasi-hereditary algebras in terms of directed boxes was given by Ovsienko.

The following theorem recently appeared in [KKO]. We do not define all the terms here, only remark that the exact structure on F(∆) is the one inherited from modB. For a gentle introduction to the theory of boxes, see [Bur].

Theorem 5.1 ([KKO], Theorem 1.1). A finite-dimensional algebra B is quasi-hereditary if and only if it is Morita equivalent to the right Burt-Butler algebraRA of a directed box A= (A, V).

Moreover, there is an equivalence of exact categories modA → F(∆).

We have the following corollary, previously obtained by Ringel by other means [Rin].

Corollary 5.2 ([KKO], Theorem 10.6). The categoryF(∆) has Auslander- Reiten sequences.

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Faculty of Professional Studies, University of Nordland, NO-8049 Bodø, Norway

E-mail address: [email protected]